| L(s) = 1 | − 2·3-s + 5-s + 9-s + 4·11-s + 2·13-s − 2·15-s + 8·17-s + 6·19-s + 4·23-s + 25-s + 4·27-s − 6·29-s − 4·31-s − 8·33-s − 10·37-s − 4·39-s + 4·41-s − 4·43-s + 45-s − 4·47-s − 16·51-s + 10·53-s + 4·55-s − 12·57-s + 14·59-s − 10·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s + 1.94·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 1.39·33-s − 1.64·37-s − 0.640·39-s + 0.624·41-s − 0.609·43-s + 0.149·45-s − 0.583·47-s − 2.24·51-s + 1.37·53-s + 0.539·55-s − 1.58·57-s + 1.82·59-s − 1.28·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.673890164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673890164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.602031191839048380886195330946, −7.47270176659216248816098497100, −6.95017797960969096171808043064, −6.06604622552007386615815319253, −5.50790923575605419739443027322, −5.08426665165117516975651321883, −3.77543296523729191688509348141, −3.18677532884258387920404839261, −1.58042090677333034067654990593, −0.877337396151041099326255800001,
0.877337396151041099326255800001, 1.58042090677333034067654990593, 3.18677532884258387920404839261, 3.77543296523729191688509348141, 5.08426665165117516975651321883, 5.50790923575605419739443027322, 6.06604622552007386615815319253, 6.95017797960969096171808043064, 7.47270176659216248816098497100, 8.602031191839048380886195330946
